EQUILIBRIUM OF RIGID BODIES 59 the chain is on the point of motion under its own weight, (1) when a is negligible compared with /, (2) when it is not negligible compared with L 11. Two equal weights are attached to the extremities of a string which hangs over a rough horizontal cylinder. Find the least amount by which either weight must be increased in order to start the system to move. The weight of the string is negligible. 12. Three cylindrical pegs of equal radius and roughness are placed at the vertices of a vertical equilateral triangle the two lower corners of which are in the same horizontal line. A string of negligible weight is attached to two weights and slung over the pegs. Find the ratio of the weights if they are on the point of motion. 13. A sphere laid upon a rough inclined plane of inclination a is on the point of sliding. Show that the coefficient of friction is I tan a. 14. A uniform ring of weight W hangs on a rough peg. A bead of weight w is fixed on the ring. Show that if the coefficient of friction W between the ring and the peg is greater than — -===-. - : the ring will be in equilibrium whatever the position of the bead with respect to the peg. 15. A uniform rod is in equilibrium with its extremities on the interior of a rough vertical hoop. Find the limiting position of the rod. 16. A weight W is suspended from the middle of a cord whose ends are attached to two rings on a horizontal pole. If w be the weight of each ring, /z the coefficient of friction, and I the length of the cord, find the greatest distance apart between the rings compatible with equilibrium.